Applied Statistics and Probability for Engineers, 7th edition 2017

15-60

15.S17 Consider the hub data in Exercise 15.S5

(a) Construct an EWMA control chart with

= 0.2 and L = 3. Comment on process control.

(b) Construct an EWMA control chart with

= 0.5 and L = 3 and compare your conclusion to part (a).

(a) The process appears to be in control.

(b) The process appears to be in control.

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15.S18 Consider a control chart for individuals with 3-sigma limits. What is the probability that there is not a signal in 3

samples? In 6 samples? In 10 samples?

The probability of having no signal is P(−3 <X< 3) = 0.9973

15.S19 The following data were considered in Quality Progress [“Digidot Plots for Process Surveillance” (1990, May, pp. 66–

68)].Measurements of center thickness (in mils) from 25 contact lenses sampled from the production process at regular

intervals are shown in the following table.

Sample

x

Sample

x

1

0.3978

14

0.3999

2

0.4019

15

0.4062

3

0.4031

16

0.4048

EWMA control chart

Applied Statistics and Probability for Engineers, 7th edition 2017

15-62

Individual chart with 3 sigma.

15.S20 A process is controlled by a P chart using samples of size 100. The center line on the chart is 0.05.

(a) What is the probability that the control chart detects a shift to 0.08 on the first sample following the shift?

(b) What is the probability that the control chart does not detect a shift to 0.08 on the first sample following the shift,

but does detect it on the second sample?

(c) Suppose that instead of a shift in the mean to 0.08, the mean shifts to 0.10. Repeat parts (a) and (b).

(d) Compare your answers for a shift to 0.08 and for a shift to 0.10. Explain why they differ. Also, explain why a shift

to 0.10 is easier to detect.

(a) The

ˆ

()P LCL UCLP

, when p = 0.08, is needed.

−−

= − = − = − →

(1 ) 0.05(1 0.05)

3 0.05 3 0.015 0

100

pp

L CL p n

Applied Statistics and Probability for Engineers, 7th edition 2017

15-63

(c) p = 0.10

15.S21 Consider the control chart for individuals with 3-sigma limits.

(a) Suppose that a shift in the process mean of magnitude

occurs. Verify that the ARL for detecting the shift is ARL =

43.9.

(b) Find the ARL for detecting a shift of magnitude 2

in the process mean.

(c) Find the ARL for detecting a shift of magnitude 3

in the process mean.

(d) Compare your responses to parts (a), (b), and (c) and explain why the ARL for detection is decreasing as the

magnitude of the shift increases.

ARL = 1/p where p is the probability a point falls outside the control limits.

(a) μ = μ0 +

and n = 1

(b) μ = μ0+ 2

Applied Statistics and Probability for Engineers, 7th edition 2017

15-64

(c) μ = μ0+ 3

15.S22 Consider an

X

control chart with UCL = 32.802, UCL = 24.642, and n = 5. Suppose that the mean shifts to 30.

(a) What is the probability that this shift is detected on the next sample?

(b) What is the ARL to detect the shift?

15.S23 The depth of a keyway is an important part quality characteristic. Samples of size n = 5 are taken every four hours from

the process, and 20 samples are summarized in the following table.

(a) Using all the data, find trial control limits for

X

and R charts. Is the process in control?

(b) Use the trial control limits from part (a) to identify out-of-control points. If necessary, revise your control limits.

Then estimate the process standard deviation.

(c) Suppose that the specifications are at 140 ± 2. Using the results from part (b), what statements can you make about

process capability? Compute estimates of the appropriate process capability ratios.

(d) To make this a 6-sigma process, the variance s2 would have to be decreased such that PCRk = 2.0. What should this

new variance value be?

Applied Statistics and Probability for Engineers, 7th edition 2017

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(e) Suppose that the mean shifts to 139.7. What is the probability that this shift is detected on the next sample? What is

the ARL after the shift?

Sample

X

r

1

139.7

1.1

2

139.8

1.4

3

140.0

1.3

4

140.1

1.6

5

139.8

0.9

6

139.9

1.0

7

139.7

1.4

8

140.2

1.2

9

139.3

1.1

10

140.7

1.0

11

138.4

0.8

12

138.5

0.9

13

137.9

1.2

14

138.5

1.1

15

140.8

1.0

16

140.5

1.3

17

139.4

1.4

18

139.9

1.0

19

137.5

1.5

20

139.2

1.3

(a)

X

and Range—Initial Study

Charting

x

X

| Range

Applied Statistics and Probability for Engineers, 7th edition 2017

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(b) Revised control limits are given in the table below:

X

and Range—Initial Study

Charting

X

Applied Statistics and Probability for Engineers, 7th edition 2017

There are no further points beyond the control limits.

2 would have to be decreased such that PCRk =

ˆ.

15-68

15.S24 The following are the number of defects observed on 15 samples of transmission units in an automotive manufacturing

company. Each lot contains five transmission units.

(a) Using all the data, compute trial control limits for a U control chart, construct the chart, and plot the data.

(b) Determine whether the process is in statistical control. If not, assume assignable causes can be found and out-of-

control points eliminated. Revise the control limits.

Sample

No. of Defects

Sample

No. of Defects

1

8

11

6

2

10

12

10

3

24

13

11

4

6

14

17

5

5

15

9

6

21

7

10

8

7

9

9

10

15

(a) The U chart follows.

Applied Statistics and Probability for Engineers, 7th edition 2017

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(b) Point 3 violates the control limits in part (b). The control limits are revised one time by omitting the out-of-control

15.S25 Suppose that the average number of defects in a unit is known to be 8. If the mean number of defects in a unit shifts to

16, what is the probability that it is detected by a U chart on the first sample following the shift

(a) if the sample size is n = 4?

(b) if the sample size is n = 10?

Use a normal approximation for U.

8u=

(a) n = 4

Applied Statistics and Probability for Engineers, 7th edition 2017

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(b) n = 10

15.S26 The number of visits (in millions) on a Web site is recorded every day. The following table shows the samples for 25

consecutive days.

(a) Estimate the process standard estimation.

(b) Set up a EMWA control chart for this process, assuming the target is 10 with

= 0.4. Does the process appear to be

in control?

Sample

Number of Visits

Sample

Number of Visits

1

10.12

16

9.66

2

9.92

17

10.42

3

9.76

18

11.30

4

9.35

19

12.53

5

9.60

20

10.76

6

8.60

21

11.92

7

10.46

22

13.24

8

10.58

23

10.64

9

9.95

24

11.31

10

9.50

25

11.26

11

11.26

26

11.79

12

10.02

27

10.53

13

10.95

28

11.82

14

8.99

29

11.47

15

9.50

30

11.76

(a) Process standard deviation is estimated using the average moving range of size 2

Applied Statistics and Probability for Engineers, 7th edition 2017

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(b) A EWMA chart with

= 0.4 and L = 3 follows. The process is not in control.

Set μ = 10 as the process mean.

15.S27 The following table shows the number of e-mails a student received each hour from 8:00 A.M. to 6:00 P.M.

The samples are collected for five days from Monday to Friday.

Hour

M

T

W

Th

F

1

2

2

2

3

1

2

2

4

0

1

2

3

2

2

2

1

2

4

4

4

3

3

2

5

1

1

2

2

1

6

1

3

2

2

1

7

3

2

1

1

0

8

2

3

2

3

1

9

1

3

3

2

0

10

2

3

2

3

0

(a) Use the rational subgrouping principle to comment on why an

X

chart that plots one point each hour with a

subgroup of size 5 is not appropriate.

(b) Construct an appropriate attribute control chart. Use all the data to find trial control limits, construct the chart, and

plot the data.

(c) Use the trial control limits from part (b) to identify out-of-control points. If necessary, revise your control limits,

assuming that any samples that plot outside the control limits can be eliminated.

Applied Statistics and Probability for Engineers, 7th edition 2017

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(b) The data are ordered sequentially so that all hours in Monday are followed by all hours in Tuesday and so forth. In

15.S28 A article in the Journal of Quality in Clinical Practice [“The Application of Statistical Process Control Charts to the

Detection and Monitoring of Hospital-Acquired Infections,” (2001, Vol. 21, pp. 112–117)] reported the use of SPC

methods to monitor hospital-acquired infections. The authors applied Shewhart and EWMA charts to the monitor

ESBL Klebsiella pneumonia infections. The monthly number of infections from June 1994 to April 1998 are shown in

the following table.

(a) What distribution might be expected for these data? What type of control chart might be appropriate?

(b) Construct the chart you selected in part (a).

Jan

Feb

Mar

April

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

1994

5

0

0

2

2

3

1

1995

1

3

2

6

4

1

2

4

3

2

8

7

1996

10

6

10

11

5

6

3

0

3

3

1

3

1997

0

2

0

4

1

1

4

2

6

7

1

5

1998

3

0

1

0

2

(c) Construct a EWMA chart for these data with

= 0.2. The article included a similarly constructed chart. What is

assumed for the distribution of the data in this chart? Can your EWMA chart perform adequately?

Applied Statistics and Probability for Engineers, 7th edition 2017

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(b) For a C chart, signals occur at points 20, 21, and 22.

(c) EWMA with

= 0.2. Signals occur at points 20–26.

Applied Statistics and Probability for Engineers, 7th edition 2017

15-74

15.S29 An article in Microelectronics Reliability [“Advanced Electronic Prognostics through System Telemetry and Pattern

Recognition Methods,” (2007, 47(12), pp. 1865–1873)] presented an example of electronic prognostics (a technique

to detect faults in order to decrease the system downtime and the number of unplanned repairs in high-reliability and

high-availability systems). Voltage signals from enterprise servers were monitored over time. The measurements are

provided in the following table.

Observation

Voltage Signal

Observation

Voltage Signal

1

1.498

26

1.510

2

1.494

27

1.521

3

1.500

28

1.507

4

1.495

29

1.493

5

1.502

30

1.499

6

1.509

31

1.509

7

1.480

32

1.491

8

1.490

33

1.478

9

1.486

34

1.495

10

1.510

35

1.482

11

1.495

36

1.488

12

1.481

37

1.480

13

1.529

38

1.519

14

1.479

39

1.486

15

1.483

40

1.517

16

1.505

41

1.517

17

1.536

42

1.490

18

1.493

43

1.495

19

1.496

44

1.545

20

1.587

45

1.501

21

1.610

46

1.503

22

1.592

47

1.486

23

1.585

48

1.473

24

1.587

49

1.502

25

1.482

50

1.497

(a) Using all the data, compute trial control limits for individual observations and moving-range charts. Construct the

chart and plot the data. Determine whether the process is in statistical control. If not, assume that assignable

causes can be found to eliminate these samples and revise the control limits.

(b) Estimate the process mean and standard deviation for the in-control process.

(c) The report in the article assumed that the signal is normally distributed with a mean of 1.5 V and a standard

deviation of 0.02 V. Do your results in part (b) support this assumption?

Applied Statistics and Probability for Engineers, 7th edition 2017

15-75

(a)

15.S30 Consider the turnaround time (TAT) for complete blood counts in Exercise 15.3.12. Suppose that the specifications for

TAT are set at 20 and 80 minutes. Use the control chart summary statistics for the following.

(a) Estimate the process standard deviation.

(b) Calculate PCR and PCRk for the process.’

Applied Statistics and Probability for Engineers, 7th edition 2017

15-76

15.S31 An article in Electric Power Systems Research [“On the Self-Scheduling of a Power Producer in Uncertain Trading

Environments” (2008, 78(3), pp. 311–317)] considered a self-scheduling approach for a power producer. The following

table shows the forecasted prices of energy for a 24-hour time period according to a base case scenario.

Hour

Price

Hour

Price

Hour

Price

1

38.77

9

48.75

17

52.07

2

37.52

10

51.18

18

51.34

3

37.07

11

51.79

19

52.55

4

35.82

12

55.22

20

53.11

5

35.04

13

53.48

21

50.88

6

35.57

14

51.34

22

52.78

7

36.23

15

45.8

23

42.16

8

38.93

16

48.14

24

42.16

(a) Construct individuals and moving-range charts. Determine whether the energy prices fluctuate in statistical control.

(b) Is the assumption of independent observations reasonable for these data?

(a) The control charts indicate the process is not in control.

15.S32 (a)

45.21,x=

21s=

. Therefore,

4

ˆ/ 21/ 0.8862 23.70sc

= = =

EWMA chart with

= 0.2 and L = 3 is shown.

Applied Statistics and Probability for Engineers, 7th edition 2017

15-77

15.S33 (a) From the individuals and moving range charts,

4.523x=

2

0.524

ˆ0.465

1.128

MR

d

= =

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